For a compact Kahler manifolds $M$ of real dimension $2n$, the Hard Lefschetz theorem states that
there is an isomorphism for each $0\leq i\leq n$,
$$ d^{n-i}\cup-\colon H^i(M;\mathbb{Q})\rightarrow H^{2n-i}(M;\mathbb{Q}),$$
where $H^i(-;\mathbb{Q})$ denotes the cohomology group with coefficients in rationals and $d\in H^2(M;\mathbb{Q})$ is the Kahler class.
My question is that is there an analog of the Hard Lefschetz theorem for quaternionic Kahler manifolds? If the answer is Yes, please send me some references, thanks ahead.
I guess the answer is yes and a possible statement may be as follows.
Let $M$ be a quaternionic Kahler manifold of real dimension $4n$, then there is a quaternionic Kahler class $d\in H^4(M;\mathbb{Q})$ and an isomorphism for each $0\leq i\leq n$: $$d^{n-i}\cup-\colon H^i(M;\mathbb{Q})\rightarrow H^{4n-3i}(M;\mathbb{Q}).$$